Question: $ C = \left[\begin{array}{rr}0 & 4 \\ 5 & 2 \\ 2 & 5\end{array}\right]$ $ B = \left[\begin{array}{rr}3 & 0 \\ 5 & 2\end{array}\right]$ What is $ C B$ ?
Solution: Because $ C$ has dimensions $(3\times2)$ and $ B$ has dimensions $(2\times2)$ , the answer matrix will have dimensions $(3\times2)$ $ C B = \left[\begin{array}{rr}{0} & {4} \\ {5} & {2} \\ \color{gray}{2} & \color{gray}{5}\end{array}\right] \left[\begin{array}{rr}{3} & \color{#DF0030}{0} \\ {5} & \color{#DF0030}{2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rr}{0}\cdot{3}+{4}\cdot{5} & ? \\ ? & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{3}+{4}\cdot{5} & ? \\ {5}\cdot{3}+{2}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rr}{0}\cdot{3}+{4}\cdot{5} & {0}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{2} \\ {5}\cdot{3}+{2}\cdot{5} & ? \\ ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{0}\cdot{3}+{4}\cdot{5} & {0}\cdot\color{#DF0030}{0}+{4}\cdot\color{#DF0030}{2} \\ {5}\cdot{3}+{2}\cdot{5} & {5}\cdot\color{#DF0030}{0}+{2}\cdot\color{#DF0030}{2} \\ \color{gray}{2}\cdot{3}+\color{gray}{5}\cdot{5} & \color{gray}{2}\cdot\color{#DF0030}{0}+\color{gray}{5}\cdot\color{#DF0030}{2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}20 & 8 \\ 25 & 4 \\ 31 & 10\end{array}\right] $